deSolve: rk4 – R documentation

Become an expert in R — Interactive courses, Cheat Sheets, certificates and more!

rk4

Solve System of ODE (Ordinary Differential Equation)s by Euler's Method or Classical Runge-Kutta 4th Order Integration.

Description

Solving initial value problems for systems of first-order ordinary differential equations (ODEs) using Euler's method or the classical Runge-Kutta 4th order integration.

Usage

euler(y, times, func, parms, verbose = FALSE, ynames = TRUE,
  dllname = NULL, initfunc = dllname, initpar = parms,
  rpar = NULL, ipar = NULL, nout = 0, outnames = NULL,
  forcings = NULL, initforc = NULL, fcontrol = NULL, ...)

rk4(y, times, func, parms, verbose = FALSE, ynames = TRUE,
  dllname = NULL, initfunc = dllname, initpar = parms,
  rpar = NULL, ipar = NULL, nout = 0, outnames = NULL,
  forcings = NULL, initforc = NULL, fcontrol = NULL, ...)

euler.1D(y, times, func, parms, nspec = NULL, dimens = NULL,
  names = NULL, verbose = FALSE, ynames = TRUE,
  dllname = NULL, initfunc = dllname, initpar = parms,
  rpar = NULL,  ipar = NULL, nout = 0, outnames = NULL,
  forcings = NULL, initforc = NULL, fcontrol = NULL, ...)

Arguments

`y`	the initial (state) values for the ODE system. If `y` has a name attribute, the names will be used to label the output matrix.
`times`	times at which explicit estimates for `y` are desired. The first value in `times` must be the initial time.
`func`	either an R-function that computes the values of the derivatives in the ODE system (the model definition) at time t, or a character string giving the name of a compiled function in a dynamically loaded shared library. If `func` is an R-function, it must be defined as: `func <- function(t, y, parms,...)`. `t` is the current time point in the integration, `y` is the current estimate of the variables in the ODE system. If the initial values `y` has a `names` attribute, the names will be available inside `func`. `parms` is a vector or list of parameters; ... (optional) are any other arguments passed to the function. The return value of `func` should be a list, whose first element is a vector containing the derivatives of `y` with respect to `time`, and whose next elements are global values that are required at each point in `times`. The derivatives must be specified in the same order as the state variables `y`. If `func` is a string, then `dllname` must give the name of the shared library (without extension) which must be loaded before `rk4` is called. See package vignette `"compiledCode"` for more details.
`parms`	vector or list of parameters used in `func`.
`nspec`	for 1D models only: the number of species (components) in the model. If `NULL`, then `dimens` should be specified.
`dimens`	for 1D models only: the number of boxes in the model. If `NULL`, then `nspec` should be specified.
`names`	for 1D models only: the names of the components; used for plotting.
`verbose`	a logical value that, when `TRUE`, triggers more verbose output from the ODE solver.
`ynames`	if `FALSE`: names of state variables are not passed to function `func` ; this may speed up the simulation especially for large models.
`dllname`	a string giving the name of the shared library (without extension) that contains all the compiled function or subroutine definitions refered to in `func`. See package vignette `"compiledCode"`.
`initfunc`	if not `NULL`, the name of the initialisation function (which initialises values of parameters), as provided in ‘dllname’. See package vignette `"compiledCode"`,
`initpar`	only when ‘dllname’ is specified and an initialisation function `initfunc` is in the DLL: the parameters passed to the initialiser, to initialise the common blocks (FORTRAN) or global variables (C, C++).
`rpar`	only when ‘dllname’ is specified: a vector with double precision values passed to the DLL-functions whose names are specified by `func` and `jacfunc`.
`ipar`	only when ‘dllname’ is specified: a vector with integer values passed to the dll-functions whose names are specified by `func` and `jacfunc`.
`nout`	only used if `dllname` is specified and the model is defined in compiled code: the number of output variables calculated in the compiled function `func`, present in the shared library. Note: it is not automatically checked whether this is indeed the number of output variables calculated in the DLL - you have to perform this check in the code. See package vignette `"compiledCode"`.
`outnames`	only used if ‘dllname’ is specified and `nout` > 0: the names of output variables calculated in the compiled function `func`, present in the shared library.
`forcings`	only used if ‘dllname’ is specified: a list with the forcing function data sets, each present as a two-columned matrix, with (time, value); interpolation outside the interval [min(`times`), max(`times`)] is done by taking the value at the closest data extreme. See forcings or package vignette `"compiledCode"`.
`initforc`	if not `NULL`, the name of the forcing function initialisation function, as provided in ‘dllname’. It MUST be present if `forcings` has been given a value. See forcings or package vignette `"compiledCode"`.
`fcontrol`	A list of control parameters for the forcing functions. See forcings or vignette `compiledCode`.
`...`	additional arguments passed to `func` allowing this to be a generic function.

Details

rk4 and euler are special versions of the two fixed step solvers with less overhead and less functionality (e.g. no interpolation and no events) compared to the generic Runge-Kutta codes called by ode resp. rk.

If you need different internal and external time steps or want to use events, please use: rk(y, times, func, parms, method = "rk4") or rk(y, times, func, parms, method = "euler").

See help pages of rk and rkMethod for details.

Function euler.1D essentially calls functioneuler but contains additional code to support plotting of 1D models, see ode.1D and plot.1D for details.

Value

A matrix of class deSolve with up to as many rows as elements in times and as many columns as elements in y plus the number of "global" values returned in the next elements of the return from func, plus and additional column for the time value. There will be a row for each element in times unless the integration routine returns with an unrecoverable error. If y has a names attribute, it will be used to label the columns of the output value.

Note

For most practical cases, solvers with flexible timestep (e.g. rk(method = "ode45") and especially solvers of the Livermore family (ODEPACK, e.g. lsoda) are superior.

Author(s)

Thomas Petzoldt thomas.petzoldt@tu-dresden.de

Examples

## =======================================================================
## Example: Analytical and numerical solutions of logistic growth
## =======================================================================

## the derivative of the logistic
logist <- function(t, x, parms) {
  with(as.list(parms), {
    dx <- r * x[1] * (1 - x[1]/K)
    list(dx)
  })
}

time  <- 0:100
N0    <- 0.1; r <- 0.5; K <- 100
parms <- c(r = r, K = K)
x <- c(N = N0)

## analytical solution
plot(time, K/(1 + (K/N0-1) * exp(-r*time)), ylim = c(0, 120),
  type = "l", col = "red", lwd = 2)

## reasonable numerical solution with rk4
time <- seq(0, 100, 2)
out <- as.data.frame(rk4(x, time, logist, parms))
points(out$time, out$N, pch = 16, col = "blue", cex = 0.5)

## same time step with euler, systematic under-estimation
time <- seq(0, 100, 2)
out <- as.data.frame(euler(x, time, logist, parms))
points(out$time, out$N, pch = 1)

## unstable result
time <- seq(0, 100, 4)
out <- as.data.frame(euler(x, time, logist, parms))
points(out$time, out$N, pch = 8, cex = 0.5)

## method with automatic time step
out <- as.data.frame(lsoda(x, time, logist, parms))
points(out$time, out$N, pch = 1, col = "green")

legend("bottomright",
  c("analytical","rk4, h=2", "euler, h=2",
    "euler, h=4", "lsoda"),
  lty = c(1, NA, NA, NA, NA), lwd = c(2, 1, 1, 1, 1),
  pch = c(NA, 16, 1, 8, 1),
  col = c("red", "blue", "black", "black", "green"))

deSolve

Solvers for Initial Value Problems of Differential Equations ('ODE', 'DAE', 'DDE')

v1.28

GPL (>= 2)

Authors

Karline Soetaert [aut] (<https://orcid.org/0000-0003-4603-7100>), Thomas Petzoldt [aut, cre] (<https://orcid.org/0000-0002-4951-6468>), R. Woodrow Setzer [aut] (<https://orcid.org/0000-0002-6709-9186>), Peter N. Brown [ctb] (files ddaspk.f, dvode.f, zvode.f), George D. Byrne [ctb] (files dvode.f, zvode.f), Ernst Hairer [ctb] (files radau5.f, radau5a), Alan C. Hindmarsh [ctb] (files ddaspk.f, dlsode.f, dvode.f, zvode.f, opdkmain.f, opdka1.f), Cleve Moler [ctb] (file dlinpck.f), Linda R. Petzold [ctb] (files ddaspk.f, dlsoda.f), Youcef Saad [ctb] (file dsparsk.f), Clement W. Ulrich [ctb] (file ddaspk.f)

Initial release

rk4